On Nelson processes with boundary condition - Project Euclid

J. Math. Soc. Japan Vol. 42, No. 2, 1990

On Nelson processes with boundary condition By Minoru YOSHIDA (Received Aug. 4, 1988) (Revised Feb. 2, 1989)

0. Introduction. In Nelson [10], it is shown that for each solution of a Schr\"odinger equation $\phi(x, t),$ $x\in R^{d},$ , with , there exists a diffusion process which has probability density at time given by , if has sufficient regularity. More precisely the following is shown in [10]: suppose that we are given a $x\in R^{d}$ , real valued function $V(x),$ $x\in R^{d}$ , and a complex valued function , and consider the Schrodinger equation with $t\geqq 0$

...

$\Vert\phi\Vert=1$

$|\phi(x, t)|^{2}$

$t$

$\phi$

$\phi_{0}(x),$

$\Vert\phi_{0}\Vert=1$

$i \frac{\partial\phi(x,t)}{\partial t}=(-\frac{1}{2}\Delta+V(x))\phi(x, t)$

,

$x\in R^{d}$

$t>0,$

, with

$\phi(x, 0)=\phi_{0}(x)$

.

For such , let $\tilde{b}(x, t)={\rm Im}\{\nabla\phi(x, f)/\phi(x, t)\}+{\rm Re}\{\nabla\phi(x, t)/\phi(x, t)\}$ . Then, under and are sufficiently regular, there exists a diffusion the assumption that process , with initial density and generator . In is such that the probability density of this diffusion at time , solves addition, in [10] it is also shown that the diffusion process the second order stochastic differential equation $(1/2)(D_{*}D+DD_{*})X_{t}=-\nabla V(X_{t})$ , are Nelson’s forward and backward stochastic derivatives. where $D$ and Generally sPeaking, the functions and may be singular, and the existence of the corresponding diffusion process must be studied carefully. Carlen $[1, 2]$ , Meyer and Zheng [8] and Nagasawa [9] considered rigorously the construction of a diffusion process with a given initial density and a generator. order to construct a Markovian propagator for the diffusion process, Carlen $[1, 2]$ used partial differential equation methods. Meyer and Zheng [8] restricted themselves and considered the construction of the to the case when $\phi(x, t)=\phi_{0}(x),$ process relationships through the between Markov processes and diffusion Dirichlet forms. We may consult Nagasawa [9] about recent developments of this area. , is a Rellich class potential In Carlen [1], it is assumed that $V(x),$ , and shown that there exists a diffusion process , which and and admits the representation at time has the probability density $\phi$

$\tilde{b}$

$\phi$

$\{X_{t}\},$

$t\geqq 0$

$(1/2)\Delta+\tilde{b}(x, t)\cdot\nabla$

$|\phi_{0}(x)|^{2}$

$t\geqq 0$

$|\phi(x, t)|^{2}$

$\{X_{t}\},$

$t\geqq 0$

$D_{*}$

$\tilde{b}$

$\phi$

$ln$

$\forall_{t\geqq 0}$

$x\in R^{d}$

$\{X_{t}\},$

$\Vert\nabla\phi_{0}\Vert^{2}